Advanced Calculus Single Variable

7.9 Exercises

Sally drives her Saturn over the 110 mile toll road in exactly 1.3 hours. The speed
limit on this toll road is 70 miles per hour and the fine for speeding is 10 dollars
per mile per hour over the speed limit. How much should Sally pay?

Two cars are careening down a freeway in Utah weaving in and out of traffic. Car
A passes car B and then car B passes car A as the driver makes obscene gestures.
This infuriates the driver of car A who passes car B while firing his handgun at
the driver of car B. Show there are at least two times when both cars have the
same speed. Then show there exists at least one time when they have the same
acceleration. The acceleration is the derivative of the velocity.

Show the cubic function f

(x)

= 5x3 + 7x − 18 has only one real zero.

Suppose f

(x)

= x7 +

|x|

+ x− 12. How many solutions are there to the equation,
f

(x)

= 0?

Let f

(x)

=

|x− 7|

+

(x − 7)

2− 2 on the interval

[6,8]

. Then f

(6)

= 0 = f

(8)

.
Does it follow from Rolle’s theorem that there exists c ∈

(6,8)

such that f′

(c)

= 0?
Explain your answer.

Suppose f and g are differentiable functions defined on ℝ. Suppose also that it is
known that

where y is some point between x and c.Fix c,x in I. Let K be a number, depending on
c,x such that

( )
∑n f(k)(c) k n+1
f (x)− f (c)+ --k!--(x− c) + K (x− c) = 0
k=1

Now the idea is to find K. To do this, let

( )
∑n f(k)(t) k n+1
F (t) = f (x)− f (t)+ k! (x− t) + K (x− t)
k=1

Then F

(x)

= F

(c)

= 0. Therefore, by Rolle’s theorem there exists y between c and x
such that F′

(y)

= 0. Do the differentiation and solve for K. This is the main result on
Taylor polynomials approximating a function f. The term f

(m+1)

(y)

(x−-c)m+1
(m+1 )!

is called
the Lagrange form of the remainder.

Let f be a real continuous function defined on the interval

[0,1]

. Also suppose f

(0)

= 0
and f

(1)

= 1 and f′

(t)

exists for all t ∈

(0,1)

. Show there exists n distinct points

{s }
i

i=1n of the interval such that

n∑ ′
f (si) = n.
i=1

Hint: Consider the mean value theorem applied to successive pairs in the following
sum.

( 1) ( 2) ( 1) ( 2)
f 3 − f (0)+ f 3 − f 3 + f (1)− f 3

Now suppose f :

[0,1]

→ ℝ is continuous and differentiable on

(0,1)

and
f

(0)

= 0 while f

(1)

= 1. Show there are distinct points

{si}

i=1n⊆

(0,1)

such
that

∑n ′ − 1
(f (si)) = n.
i=1

Hint: Let 0 = t0< t1<

⋅⋅⋅

< tn = 1 and pick xi∈ f−1

(t )
i

such that these
xi are increasing and xn = 1,x0 = 0. Explain why you can do this. Then
argue

ti+1 − ti = f (xi+1)− f (xi) = f′(si)(xi+1 − xi)

and so

xi+1-−-xi --1--
ti+1 − ti = f′(si)

Now choose the ti to be equally spaced.

Show that

(x + 1)

3∕2− x3∕2> 2 for all x ≥ 2. Explain why for n a natural
number larger than or equal to 1, there exists a natural number m such that

(n + 1)

3> m2> n3. Hint:Verify directly for n = 1 and use the above inequality to
take care of the case where n ≥ 2. This shows that between the cubes of any two natural
numbers there is the square of a natural number.